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Let 
$\Omega \subset \mathbb{R}^d$ with 
$d\geq 2$ be a bounded domain of class 
${\mathcal C}^{1,\beta }$ for some 
$\beta \in (0,1)$. For 
$p\in (1, \infty )$ and 
$s\in (0,1)$, let 
$\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator 
$-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for 
$\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of 
$\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for 
$-\Delta _p+(-\Delta _p)^s$.
